{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:04:45Z","timestamp":1760148285999,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,16]],"date-time":"2023-04-16T00:00:00Z","timestamp":1681603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Technology, Taiwan","award":["109-2221-E-155-035-MY3","110-2221-E-A49-024-MY3"],"award-info":[{"award-number":["109-2221-E-155-035-MY3","110-2221-E-A49-024-MY3"]}]},{"name":"Ministry of Science and Technology, Taiwan","award":["109-2221-E-155-035-MY3","110-2221-E-A49-024-MY3"],"award-info":[{"award-number":["109-2221-E-155-035-MY3","110-2221-E-A49-024-MY3"]}]},{"name":"Natural Sciences and Engineering Research Council (NSERC) of Canada","award":["109-2221-E-155-035-MY3","110-2221-E-A49-024-MY3"],"award-info":[{"award-number":["109-2221-E-155-035-MY3","110-2221-E-A49-024-MY3"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The error probability of block codes sent under a non-uniform input distribution over the memoryless binary symmetric channel (BSC) and decoded via the maximum a posteriori (MAP) decoding rule is investigated. It is proved that the ratio of the probability of MAP decoder ties to the probability of error grows most linearly in blocklength when no MAP decoding ties occur, thus showing that decoder ties do not affect the code\u2019s error exponent. This result generalizes a similar recent result shown for the case of block codes transmitted over the BSC under a uniform input distribution.<\/jats:p>","DOI":"10.3390\/e25040668","type":"journal-article","created":{"date-parts":[[2023,4,17]],"date-time":"2023-04-17T02:26:02Z","timestamp":1681698362000},"page":"668","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Decoder Ties for the Binary Symmetric Channel with Arbitrarily Distributed Input"],"prefix":"10.3390","volume":"25","author":[{"given":"Ling-Hua","family":"Chang","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Yuan Ze University, Taoyuan 32003, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1231-0706","authenticated-orcid":false,"given":"Po-Ning","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Communications Engineering, National Yang-Ming Chiao-Tung University, Taipei 112304, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7980-724X","authenticated-orcid":false,"given":"Fady","family":"Alajaji","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Queen\u2019s University, Kingston, ON K7L 3N6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3501","DOI":"10.1109\/TIT.2022.3150597","article-title":"Decoder Ties Do Not Affect the Error Exponent of the Memoryless Binary Symmetric Channel","volume":"68","author":"Chang","year":"2022","journal-title":"IEEE Trans. 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